ON THE CARLEMAN FORMULA IN SIEGEL DOMAINS

Abstract

This paper investigates the Carleman formulas within the framework of Siegel domains, which are prominent structures in the theory of several complex variables. Siegel domains, particularly of the second kind, provide a natural setting for the study of holomorphic functions and their boundary behaviors due to their geometric and analytical properties. The Carleman formula serves as a reproducing integral formula, enabling the analytic continuation of functions from boundary subsets to the interior of the domain. In this study, we present generalized Carleman-type formulas adapted to the geometry of Siegel domains, incorporating Hermitian forms and differential structures. Special attention is given to the construction of integral kernels and the application of these formulas to classes of holomorphic functions The results demonstrate how function values in a Siegel domain can be recovered from their boundary data, and how complex lines and automorphisms play a crucial role in the formulation. This contributes to a deeper understanding of boundary integral methods and the role of geometric symmetry in complex analysis.